Problem: Let $a(x)=5x^3-3x^2+2x+6$, and $b(x)=x^3$. When dividing $a$ by $b$, we can find the unique quotient polynomial $q$ and remainder polynomial $r$ that satisfy the following equation: $\dfrac{a(x)}{b(x)}=q(x) + \dfrac{r(x)}{b(x)}$, where the degree of $r(x)$ is less than the degree of $b(x)$. What is the quotient, $q(x)$ ? $ q(x)=$ What is the remainder, $r(x)$ ? $r(x)=$
Solution: Note that $a(x)$ has the same degree as $b(x)$. This allows us to find a non-zero quotient polynomial, $q(x)$. [Why is this important?] Let's rewrite the fraction to cancel common factors: $ \begin{aligned} \dfrac{a(x)}{b(x)}=\dfrac{5x^3-3x^2+2x+6}{x^3}&=\dfrac{5 {x^3}}{ {x^3}}+\dfrac{-3x^2+2x+6}{x^3}\\\\ &={5}+\dfrac{{-3x^2+2x+6}}{x^3}\\\\ &={q(x)} + \dfrac{{r(x)}}{b(x)}\end{aligned}$ Since the degree of ${-3x^2+2x+6}$ is less than the degree of $x^3$, it follows that ${r(x)}={-3x^2+2x+6}$, and ${q(x)}={5}$. To conclude, $q(x)=5$ $r(x)=-3x^2+2x+6$ [Is there another way of doing this?]